Full transcript of “Lecture 3.3 - Describing Numerical Data - Median and Mode”. Click a timestamp to open the video at that moment. Need another video? Use the YouTube transcript generator.
0:14 This is what we another popularly or frequently used measure is what we refer to as a median. Essentially the median of a data set is the number that divides the data set into a bottom 50 percent and a top 50 percent. The minute we say top 50 percent and a bottom 50 percent, we quickly realize that we are talking about an ordered data set. So, I formally define a median of a data set is the middle value in the ordered list. The ordered list is extremely important here. So, now how do I compute the median of a data set? The computing data, so I have an ordered list. So, I arrange the data in increasing
1:07 order. Let n denote the number of observations in the data set. Now, this is important, if the number of observations is odd, then the median of is exactly in the middle of the ordered list. For example, if my observations x 1 x 2 x 3 x 4 x 5, 5 n equal to 5, 5 is odd, then this is assume it is ordered. So, for order data, let me introduce a notation x 1 x 2 x 3 x 4 and x 5, x 1 is the order that is the first data, x 2 is the second, x 3, this is my ordered data, then the data in the n
1:58 plus 1 by 2, n plus 1 by 2 is 5 plus 1 - 6 divided by 2, third. So, this would be my median. Remember x 1, x 2, x 3, x 4, x 5 are are is my data arranged in increasing order. And x 3 which is the third rank data would be my median. If the number of observations is even, for example, I have x 6 also in this case, my n equal to 6, then the median is going to be your x 3 that is my n by 2 observation
2:44 plus n by 2 plus 1 x 4 divided by 2, that is what this means. So, if at the median depends on whether the number of observation is an odd number or a even number. So, now let us apply this definition and steps to compute the median for the data sets we have already seen before. So, when I have this data 2, 12, 5, 7, 6 and 3, the first step says arrange the data in increasing order. So, the arrangement of data is going to be 2, 3, 2, 3, 5, 6, 7, 7, and 12. I have arranged my data in ascending order.
3:39 Now, what is my n in this case? I have 1, 2, 3, 4, 5, 6, 7, n equal to 7 which is odd. So, if n equal to odd, then I apply my, first n equal to odd, the median is n plus 1 by 2. So, I have n equal to 7 which is odd. So, median is 8 by 2 which is the 4th observation which is equal to 6. So, my median of this data set is 6. Now, let us look at another example, again the same example. Remember when we are looking at the same examples which we computed the mean for. The difference between the second data set and the first data set is in only one observation which is the second observation
4:33 here which is 105 for the second data set, and 12 for the first data set. Remember when we computed the mean, we saw that this one observation actually influenced the mean, and the mean of the first data set and the second data set were very different from each other. Now, let us see what happens to the median of these two data sets. The number of observations again is the same. I arrange the data in ascending order. So, when I arrange this data in ascending order, I have 2, I have 3, I have 5, 6, 7, 7, and I have a 105. The number of data is 7 which is again odd. The median is again
5:20 the 4th observation and which is again 6; it does not change. The median is the 4th observation which does not change which is equal to 6. So, what you can immediately notice here is while the mean was very different for these two data sets, the median is the same for both the data sets even though it differs very drastically in an outlier. So, the median is not very sensitive to the outliers the way the mean was. Now, let us look at the third data set which had only 6 observations. Again I arrange this data in ascending order. So, I have a 2, I
6:06 have a 3, I have a 5, 6, 7 and 105. Again I have n which is equal to 6. So, my median is going to be the mean of the 3rd observation and the 4th observation which is 5 plus 6 by 2 which would give me 5.5. Notice that this 5.5 is not a member of the data set. So, the median need not belong to the data set. Whereas, for the first two data, the date median was a member because we are stay looking at a particular observation; whereas, here I am not looking at a particular observation.
6:53 So, let us go back to our example here. So, for here the sample means were the same. In the second one, the sample mean is 21, the sample median is 6. The first data set and second data set only differ in the second observation which is 117 here and 12 here. And we already see that there is a significant difference in the mean, but the median remains the same. So, the sample mean is sensitive to outliers, whereas the sample median is not sensitive to outliers. Now, let us compute in our Google sheet. What
7:39 happens in our Google sheet? So, what is the median? So, you can see that in the Google sheet, the median of data is obtained by the function median B 2 to B 11 which will give me the data. So, now I can see that the median here is 63.5. How did we obtain 63.5? I can arrange this data. So, if I arrange this data, n equal to even; n equal to 10, which is equal to even. Sorry, 1, 2, 3, 4, 61 plus 66, it is
8:31 which is 63.5. And that is what I have here which is 63.5, because 61 is my 5th 5th ranked observation here. So, this is 5th 61 plus 66, I have these observations here. So, you can see the 61 plus 66 which is 137 divided by 2 which will give me 63.5 which is the median. I can get this through the command median of the array in Google sheets. So, what happens when we add a constant to the data set? When I add a constant to the data set, again let y i be x i plus a constant,
9:19 where c is a constant, then what happens to my new median? So, let us go back to the example 68, 79, these are the marks of my students. Again I arrange them in ascending order. When I arrange them in ascending order, we saw that the mean is 61 plus 66 by 2 which is 63.5. Again if I decide and the teacher decides to add 5 marks to every student, then the data becomes 40, 43, 47 plus 5 which is 52, 63, 66, 73, 75, and 84; I am adding 5 to each point of the data set. Now, again you notice
10:07 that by adding a constant to the data set does not change the order of the observation. So, here this n by 2 observation was 61, and n by 2 plus 1 is 66. So, corresponding to 61, I have 66; corresponding to 66 I have 71. The n does not change; the number of observations does not change. So, the median in this case is 66 plus 71 divided by 2, and you can see that it is 68.5. Whereas, 66 is was 61 plus 5 - 66 was 71, 66 plus 5. So, the new median
10:58 is nothing but your old median plus a constant, because the values are the same. If I have x, so here it is x 5 plus x 6 by 2 was my old median. My y 5 is y 5 plus y 6 by 2 is my new median. But y 5 was x 5 plus my constant, y 6 is x 6 plus my constant which is 5. So, I have the new median is x 5 plus x 6 by 2 which is my old median plus 5. So, 5 is the constant. Old median plus 5 is my new
11:45 median. So, whenever I am adding a constant, the new median is your old median plus the constant. It does not it it you are adding that constant to the new median. What happens when you multiply the entire data set with a constant? When I multiply the data set with an entire constant, again my old data set, so y 1, so I had y 5 plus y 6 by 2 which is my new median so, but y 5 is x y plus into c, y 6 is going to be x 6 into c, I can remove the c. So, I have c into x 5 plus x 6 by 2, x 5 plus x 6 by 2
12:36 was my old median. So, my old median into the constant will give me the new median. And this we can see in our example. So, again recall we know the median is 63.5. If I scale down or each mark is multiplied by 0.4, I saw that this is my what is happening to my data set. Again I arrange the data set in ascending order, my 5th observation here is 24.4, 6th observation is 26.4, the median of the new data set hence is 25.4, which I can verify is 0.4 into 63.5; 25.4 is the new median which is the old median into 63.5 .
13:30 So, when I go back and see that in my Google sheets, so when I am add a constant, so you can see that the old median is 63.5, when I add a constant of 5, 63.5 plus 5 is 68.5, whereas 63.5 into 0.4 gives me 25.4. So, this is how we can obtain our new median. Now, we move on to the third measure of central tendency which we refer to as a mode. We have already seen what is a mode while describing categorical data. We see that the mode as we defined when we talked about categorical data is that observation which has the highest
14:19 frequency of occurrence. So, that is the same way we define even for numerical data. So, the mode of a data set as it is given here is the most frequency frequently occurring value, so that is what we refer to as a mode. So, now how do we obtain a mode? Just as we did in the case of the categorical data, even in the numerical data, what we do is we check for the mode by computing or calculating that observation which appears the most number of ties. If a value occurs more than once, if no value occurs more than one, the data set has no mode; otherwise that value which
15:05 occurs with the greatest frequency is the mode of a data set. So, now moving forward we find again we go back to the same data sets which you have considered so far. In this data set I have 2, 12, 5, 7, 7, 6, 3. We can see that 7 appears twice, hence the mode of this data set is 7. The second data set also 7 occurs twice, again you can see that the difference between the first data set and the second data set is only one observation, namely 12 appears in
15:51 the first data set, 105 appears in the second data set. The mode again is 7 for this data set. The third data set has all 6 values that are distinct; hence there is no mode for the third data set. Now, again if you look at the first and second data set, recall when we computed the mean, the mean was very different for both these data sets, the median was the same, the mode is also the same for both the data sets. Now, as we did in the case of the mean and the median, let us see what happens when we manipulate a data set namely when we add a
16:36 constant and when we multiply with a constant. When we add a constant to each of the observations of the data set, for example, I have 1, 2, 3, and I am adding a constant to each one of them, this becomes 3, this becomes 4, and this becomes 5, I can see that nothing the the characteristic of the data set remains the same. So, the mode of the data set the new mode is just the old mode plus c. So, the new mode is that. So, if I have x 1 x 2 x 3 which is my old data set, and suppose x 2 is the mode of my old data set, I add to get y 1, y 2 y 3 which is y x 1 plus a constant x 2 plus
17:31 a constant, and x 3 plus a constant. If x x 2 is that which appears the most number of time, so it would be x 1 x 2 x 1 x 2 x 2 x 3, x 2 is the mode. So, it becomes y 1 y 2 y 2 y 3 where y 2 appears the most number of times which so you can see that the new mode is y x 2 plus c, hence the new mode of my data set is old mode plus the constant. Recall again going back to our marks example the data set is 68, 79. I have 10 students,
18:16 and I can see that the mode for this data set is 68. I add 5 marks just as earlier. And you can see that the data set in ascending order becomes 40, 43, I do not need it in ascending order now, but nevertheless I can see that the mode now is 73 which corresponds to 68 plus 5. Hence the new mode is nothing but the old mode plus the constant. What happens when we multiply a constant? When we multiply a constant again the new mode is nothing but the old mode times the constant. Again the reasoning is very simple. Suppose, I have a data set x 1 x 2 x 2 x 3, x 2 being the mode here; I have y 1 y 2 y
19:09 2 y 3, where y 1 is x 1 into a constant, y 2 is the mode here. And I know y 2 is nothing but x 2 times the constant; x 2 is the mode for my earlier data set. So, the new mode is old mode into the constant. So, again recall the example the mode is 68, this is what we saw from the earlier table example. Now, if the teacher decides to scale down each mark by 40 percent and each mark is multiplied by 0.4, the data set becomes the following 27.2, 31.2. The new mode is 27.2, this appears twice. And we can verify
19:55 that this 27.2 is 0.4 times 68. So, if we look at this, we can go back to our numerical summaries. So, you can see that this is my data set. This I arrange my data set in ascending order here. This is the data set. The highlighted portion is the data set in my Google sheets. So, you can see from this 68 is that value that appears twice, and that is given by the function mode - m o d e times the data returns the value 68. When I add a constant, the mode of the new data set is 73; 73 is 68 plus 5.
20:51 And 27.2 is when I multiply it with a constant 27.2 is 68 times 0.4, so that is what we have seen. So, moving forward what we have seen so far is we have studied about the measures of central tendency, namely we looked at what is the mean, we define both the population mean and the sample mean. But then our discussion centered mostly around the sample mean. Then we moved on to define what is a median of a data set, and then what is a mode of a data set. For each one of these operations or measures, we saw what was the impact of adding a constant or multiplying
21:38 with a constant on the measures. That is what we have.
